Algebraic Characterizations of Complexity-Theoretic Classes of Real FunctionsReport as inadecuate

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1 LIX - Laboratoire d-informatique de l-École polytechnique Palaiseau 2 Faculty of Engineering, Alexandria University 3 CARTE - Theoretical adverse computations, and safety Inria Nancy - Grand Est, LORIA - FM - Department of Formal Methods

Abstract : Recursive analysis is the most classical approach to model and discuss computations over the real numbers.Recently, it has been shown that computability classes of functions in the sense of recursive analysis can be defined or characterized in an algebraic machine independent way, without resorting to Turing machines. In particular nice connections between the class of computable functions and some of its sub- and sup-classes over the reals and algebraically defined sub- and sup- classes of R-recursive functions à la Moore 96 have been obtained. However, until now, this has been done only at the computability level, and not at the complexity level. In this paper we provide a framework that allows us to dive into the complexity level of real functions. In particular we provide the first algebraic characterization of polynomial-time computable functions over the reals. This framework opens the field of implicit complexity of analog functions, and also provides a new reading of some of the existing characterizations at the computability level.

Keywords : Recursive Analysis Polynomial Time Algebraic Characterization Real Computation Oracle Turing Machines

Author: Olivier Bournez - Walid Gomaa - Emmanuel Hainry -



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